Found problems: 233
Let the sequence $a_i$ be defined as $a_{i+1} = 2^{a_i}$. Find the number of integers $1 \le n \le 1000$ such that if $a_0 = n$, then $100$ divides $a_{1000} - a_1$.
Suppose $P (x)$ is a polynomial with real coefficients such that $P (t) = P (1)t^2 + P (P (1))t + P (P (P (1)))$ for all real numbers $t$. Compute the largest possible value of $P(P(P(P(1))))$.
During the weekends, Eli delivers milk in the complex plane. On Saturday, he begins at $z$ and delivers milk to houses located at $z^3,z^5,z^7,\ldots,z^{2013}$ in that order; on Sunday, he begins at $1$ and delivers milk to houses located at $z^2,z^4,z^6,\ldots,z^{2012}$ in that order. Eli always walks directly (in a straight line) between two houses. If the distance he must travel from his starting point to the last house is $\sqrt{2012}$ on both days, find the real part of $z^2$.
Seven lattice points form a convex heptagon with all sides having distinct lengths. Find the minimum possible value of the sum of the squares of the sides of the heptagon.
We have $10$ points on a line $A_1,A_2\ldots A_{10}$ in that order. Initially there are $n$ chips on point $A_1$. Now we are allowed to perform two types of moves. Take two chips on $A_i$, remove them and place one chip on $A_{i+1}$, or take two chips on $A_{i+1}$, remove them, and place a chip on $A_{i+2}$ and $A_i$ . Find the minimum possible value of $n$ such that it is possible to get a chip on $A_{10}$ through a sequence of moves.
Let $R$ be the region in the Cartesian plane of points $(x,y)$ satisfying $x\geq 0$, $y\geq 0$, and $x+y+\lfloor x\rfloor+\lfloor y\rfloor\leq 5$. Determine the area of $R$.
Let $W$ be the hypercube $\{(x_1,x_2,x_3,x_4)\,|\,0\leq x_1,x_2,x_3,x_4\leq 1\}$. The intersection of $W$ and a hyperplane parallel to $x_1+x_2+x_3+x_4=0$ is a non-degenerate $3$-dimensional polyhedron. What is the maximum number of faces of this polyhedron?
Let $S = \{a_1, \ldots, a_n \}$ be a finite set of positive integers of size $n \ge 1$, and let $T$ be the set of all positive integers that can be expressed as sums of perfect powers (including $1$) of distinct numbers in $S$, meaning
\[ T = \left\{ \sum_{i=1}^n a_i^{e_i} \mid e_1, e_2, \dots, e_n \ge 0 \right\}. \]
Show that there is a positive integer $N$ (only depending on $n$) such that $T$ contains no arithmetic progression of length $N$.
[i]Yang Liu[/i]
You are given an unlimited supply of red, blue, and yellow cards to form a hand. Each card has a point value and your score is the sum of the point values of those cards. The point values are as follows: the value of each red card is 1, the value of each blue card is equal to twice the number of red cards, and the value of each yellow card is equal to three times the number of blue cards. What is the maximum score you can get with fifteen cards?
Alice writes $1001$ letters on a blackboard, each one chosen independently and uniformly at random from the set $S=\{a, b, c\}$. A move consists of erasing two distinct letters from the board and replacing them with the third letter in $S$. What is the probability that Alice can perform a sequence of moves which results in one letter remaining on the blackboard?
[i]Proposed by Daniel Zhu.[/i]
Let $A_1,A_2,\ldots,A_m$ be finite sets of size $2012$ and let $B_1,B_2,\ldots,B_m$ be finite sets of size $2013$ such that $A_i\cap B_j=\emptyset$ if and only if $i=j$. Find the maximum value of $m$.
Let $x_1=y_1=x_2=y_2=1$, then for $n\geq 3$ let $x_n=x_{n-1}y_{n-2}+x_{n-2}y_{n-1}$ and $y_n=y_{n-1}y_{n-2}-x_{n-1}x_{n-2}$. What are the last two digits of $|x_{2012}|?$
Circles $C_1$, $C_2$, $C_3$ have radius $ 1$ and centers $O, P, Q$ respectively. $C_1$ and $C_2$ intersect at $A$, $C_2$ and $C_3$ intersect at $B$, $C_3$ and $C_1$ intersect at $C$, in such a way that $\angle APB = 60^o$ , $\angle BQC = 36^o$ , and $\angle COA = 72^o$ . Find angle $\angle ABC$ (degrees).
Given $w$ and $z$ are complex numbers such that $|w+z|=1$ and $|w^2+z^2|=14$, find the smallest possible value of $|w^3+z^3|$. Here $| \cdot |$ denotes the absolute value of a complex number, given by $|a+bi|=\sqrt{a^2+b^2}$ whenever $a$ and $b$ are real numbers.
Triangle $ ABC$ has $ \angle C \equal{} 60^{\circ}$ and $ BC \equal{} 4$. Point $ D$ is the midpoint of $ BC$. What is the largest possible value of $ \tan{\angle BAD}$?
$ \textbf{(A)} \ \frac {\sqrt {3}}{6} \qquad \textbf{(B)} \ \frac {\sqrt {3}}{3} \qquad \textbf{(C)} \ \frac {\sqrt {3}}{2\sqrt {2}} \qquad \textbf{(D)} \ \frac {\sqrt {3}}{4\sqrt {2} \minus{} 3} \qquad \textbf{(E)}\ 1$
A [i]root of unity[/i] is a complex number that is a solution to $ z^n \equal{} 1$ for some positive integer $ n$. Determine the number of roots of unity that are also roots of $ z^2 \plus{} az \plus{} b \equal{} 0$ for some integers $ a$ and $ b$.
We have $10$ points on a line $A_1,A_2\ldots A_{10}$ in that order. Initially there are $n$ chips on point $A_1$. Now we are allowed to perform two types of moves. Take two chips on $A_i$, remove them and place one chip on $A_{i+1}$, or take two chips on $A_{i+1}$, remove them, and place a chip on $A_{i+2}$ and $A_i$ . Find the minimum possible value of $n$ such that it is possible to get a chip on $A_{10}$ through a sequence of moves.
Find the number of triples of sets $(A, B, C)$ such that:
(a) $A, B, C \subseteq \{1, 2, 3, \dots , 8 \}$.
(b) $|A \cap B| = |B \cap C| = |C \cap A| = 2$.
(c) $|A| = |B| = |C| = 4$.
Here, $|S|$ denotes the number of elements in the set $S$.
In rectangle $ABCD$, points $E$ and $F$ lie on sides $AB$ and $CD$ respectively such that both $AF$ and $CE$ are perpendicular to diagonal $BD$. Given that $BF$ and $DE$ separate $ABCD$ into three polygons with equal area, and that $EF = 1$, find the length of $BD$.
The numbers $1, 2\ldots11$ are arranged in a line from left to right in a random order. It is observed that the middle number is larger than exactly one number to its left. Find the probability that it is larger than exactly one number to its right.
Two concentric circles have radii $1$ and $2$. Two points on the outer circle are chosen independently and uniformly at random. What is the probability that the chord joining the two points intersects the inner circle?
$\textbf{(A) }\frac{1}{6}\qquad\textbf{(B) }\frac{1}{4}\qquad\textbf{(C) }\frac{2-\sqrt{2}}{2}\qquad\textbf{(D) }\frac{1}{3}\qquad\textbf{(E) }\frac{1}{2}\qquad$
Seven lattice points form a convex heptagon with all sides having distinct lengths. Find the minimum possible value of the sum of the squares of the sides of the heptagon.
Convex hexagon $ABCDEF$ is drawn in the plane such that $ACDF$ and $ABDE$ are parallelograms with area 168. $AC$ and $BD$ intersect at $G$. Given that the area of $AGB$ is 10 more than the area of $CGB$, find the smallest possible area of hexagon $ABCDEF$.
DeAndre Jordan shoots free throws that are worth $1$ point each. He makes $40\%$ of his shots. If he takes two shots find the probability that he scores at least $1$ point.
The lines $y=x$, $y=2x$, and $y=3x$ are the three medians of a triangle with perimeter $1$. Find the length of the longest side of the triangle.